\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 49, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/49\hfil Multiple positive periodic solutions]
{Multiple positive periodic solutions to a non-autonomous 
 Lotka-Volterra predator-prey system with harvesting terms}

\author[Kaihong Zhao, Yongkun Li \hfil EJDE-2011/49\hfilneg]
{Kaihong Zhao, Yongkun Li} 

\address{Kaihong Zhao \newline
Department of Applied Mathematics\\
Kunming University of Science and Technology\\
Kunming, Yunnan 650093, China}
\email{zhaokaihongs@126.com}

\address{Yongkun Li \newline
Department of Mathematics, Yunnan University\\
Kunming, Yunnan 650091, China}
\email{yklie@ynu.edu.cn}

\thanks{Submitted November 26, 2010. Published April 7, 2011.}
\thanks{Supported by grant 10971183 from the National Natural
Sciences Foundation, China}
\subjclass[2000]{34C25, 92D25}
\keywords{Periodic solutions;  Lotka-Volterra network;
 predator-prey system; \hfill\break\indent
 coincidence  degree; harvesting term}

\begin{abstract}
 Using Mawhin's continuation theorem of coincidence degree
 theory, we establish the existence  of $2^{n+m}$ positive 
 periodic solutions for a  non-autonomous Lotka-Volterra 
 network-like predator-prey  system with harvesting terms. 
 Here $n$ and $m$ denote the number of prey and predator species 
 respectively. An example is given to illustrate our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and description of the model}

 In the usual predator-prey model, there is only one predator
and one prey. However, in nature we encounter complex systems with
several species as predators and several species as prey. In our model
all predators form one layer, and all prey form another layer; to be
called predator layer and prey layer, respectively.
 There is a competition relationship among each species lying
in the same layer because they fight for food, living space and so on.
Considering the above, in this paper, we introduce the following
non-autonomous Lotka-Volterra network-like predator-prey system
with harvesting terms
\begin{equation} \label{e1.1}
  \begin{gathered}
\begin{aligned}
 \dot{x}_i(t)&=x_i(t)\Big(a_i(t)-b_i(t)x_i(t)
-\sum_{r=1,r\neq i}^{n}c_{ir}(t)x_{r}(t)\\
    &\quad-\sum_{k=1}^{m}d_{ik}(t)x_{n+k}(t)\Big)-h_i(t), \quad
 i=1,2,\dots,n,
\end{aligned}\\
\begin{aligned}
 \dot{x}_{n+j}(t)
&= x_{n+j}(t)\Big(\alpha_j(t)-\beta_j(t)x_{n+j}(t)
    -\sum_{r=1,r\neq j}^{m}\gamma_{rj}(t)x_{n+r}(t)\\
&\quad +\sum_{k=1}^{n}\delta_{kj}(t)x_{k}(t)\Big)-e_j(t),\quad
  j=1,2,\dots,m
  \end{aligned}
\end{gathered}
\end{equation}
where $x_i(t)$ and $x_{n+j}(t)$ $(j=1,2,\dots,m)$ are the $i$th prey
species population density and the $j$th predator species population
density, respectively; $a_i(t), b_i(t)$ and $h_i(t)$ stand for
the $i$th prey species birth rate, death rate and harvesting rate,
respectively; $\alpha_j(t),\beta_j(t)$ and $e_j(t)$ stand for
the $j$th predator species birth rate, death rate and harvesting
rate, respectively; $c_{is}(t)(i\neq s)$ represents the competition
rate between the $s$th prey species and the $i$th prey species,
$d_{ik}(t)$ $(i=1,2,\dots,n;k=1,2,\dots,m)$ represents the $k$th
predator species predation rate on the $i$th prey species,
$\gamma_{sj}(t)(s\neq j)$ stands for the competition rate between
the $s$th predator species and the $j$th predator species,
$\delta_{kj}(t)$ $(j=1,2,\dots,m;k=1,2,\dots,n)$ stands for the
transformation rate between the $k$th prey species and the $n+j$th
predator species. In addition, the effects of a periodically varying
environment are important for evolutionary theory as the selective
forces on systems in a fluctuating environment differ from those in
a stable environment. Therefore, the assumptions of periodicity of
the parameters are a way of incorporating the periodicity of the
environment (e.g, seasonal effects of weather, food supplies, mating
habits, etc), which leads us to assume that
$a_i(t),b_i(t),c_{is}(t),d_{ik}(t),h_i(t),\alpha_j(t),\beta_j(t),\gamma_{sj}(t),\delta_{kj}(t)$
and $e_j(t)$ $(i=1,2,\dots,n;j=1,2,\dots,m)$ are all positive
continuous $\omega$-periodic functions.

Since a very basic and important problem in the study of a
population growth model with a periodic environment is the global
existence and stability of a positive periodic solution, which plays
a similar role as a globally stable equilibrium does in an
autonomous model. This motivates us to investigate the existence of
a positive periodic or multiple positive periodic solutions for
system \eqref{e1.1}. In fact, it is more likely for some biological species
to take on multiple periodic change regulations and have multiple
local stable periodic phenomena. Therefore, it is essential  for us
to investigate the existence of multiple positive periodic solutions
for population models. Our main purpose of this paper is by using
Mawhin's continuation theorem of coincidence degree theory
\cite{g1}, to establish the existence of $2^{n+m}$ positive periodic solutions for
system \eqref{e1.1}. For the work concerning the multiple existence of
periodic solutions of periodic population models which was done by
using coincidence degree theory, we refer the reader to
\cite{c1,f1,h1,w1,x1}.

 The organization of
the rest of this paper is as follows. In Section 2, by employing the
continuation theorem of coincidence degree theory and the skills of
inequalities, we establish the existence of $2^{n+m}$ positive
periodic solutions of system \eqref{e1.1}. In Section 3, one example is
given to illustrate the effectiveness of our results.


\section{Existence of $2^{n+m}$ positive periodic solutions}


In this section, by using Mawhin's continuation theorem and some
inequalities, we shall show the existence of positive periodic
solutions of \eqref{e1.1}. To do so, we need to make some preparations.

Let $X$ and $Z$ be real normed vector spaces.
Let $L:\operatorname{Dom} L\subset X\to Z$ be a linear mapping and
$N: X\times [0,1] \to Z$ be a continuous mapping.
The mapping $L$ will be
called a Fredholm mapping of index zero if dim
$\ker L = \operatorname{codim}\operatorname{Im}L<\infty$
 and $\operatorname{Im} L$ is closed in $Z$.
If $L$ is a Fredholm mapping
of index zero, then there exists continuous projectors
$P:X\to X$ and $Q:Z\to Z$ such that $\operatorname{Im}
P=\ker L$ and $\ker  Q=\operatorname{Im}L=\operatorname{Im}(I-Q)$,
and $X=\ker L \oplus
\ker  P, Z=\operatorname{Im} L \oplus \operatorname{Im} Q$. It
follows that
$L|_{\operatorname{Dom} L \cap \ker P}:(I-P)X\to \operatorname{Im} L$
is invertible and its inverse is denoted by $K_P$.
If $\Omega$ is a bounded open subset of $X$,
the mapping $N$ is called $L$-compact on $\bar{\Omega}\times [0,1]$,
if $QN(\bar{\Omega}\times [0,1])$ is bounded and
$K_P(I-Q)N:\bar{\Omega}\times [0,1]\to X$ is compact.
Because $\operatorname{Im}Q$ is isomorphic to $\ker L$,
there exists an isomorphism
$J: \operatorname{Im} Q\to \ker L$.

The Mawhin's continuous theorem \cite[p.40]{g1} is  as follows.

\begin{lemma}[\cite{g1}] \label{lem2.1}
Let $L$ be a Fredholm mapping of index zero and let $N$ be
$L$-compact on $\bar{\Omega}\times [0,1]$.
 Assume
\begin{itemize}
\item[(a)] for each $\lambda\in (0,1)$, every solution $x$ of $L
x=\lambda N(x,\lambda)$ is such that $x\notin \partial \Omega\cap
\operatorname{Dom} L$;
\item[(b)] $QN(x,0)x\neq 0$ for each $x\in \partial\Omega\cap
\ker  L$;
\item[(c)] $\deg(JQN(x,0),\Omega\cap \ker  L,0)\neq 0$.
\end{itemize}
Then $Lx=Nx$ has at least one solution in $\overline{\Omega}\cap
\operatorname{Dom} L$.
\end{lemma}

For the sake of convenience, we denote
$f^{l}=\min_{t\in[0,\omega]}f(t)$,
$f^{M}=\max_{t\in[0,\omega]}f(t)$,
$\bar{f}=\frac{1}{\omega}\int^{\omega}_{0}f(t)\,\mathrm{d}t$,
respectively, here $f(t)$ is a continuous $\omega$-periodic
function.

For simplicity, we need to introduce some notations as follows.
\begin{gather*}
l_i^{\pm}=\frac{a_i^{M}\pm\sqrt{(a_i^{M})^{2}
-4b_i^{l}h_i^{l}}}{2b_i^{l}},
\\
l_{n+j}^{\pm}=\frac{(\alpha_j^{M}
+\sum_{k=1}^{n}\delta_{kj}^{M}l_{k}^{+})\pm\sqrt{(\alpha_j^{M}
+\sum_{k=1}^{n}\delta_{kj}^{M}l_{k}^{+})^{2}-4\beta_j^{l}
e_j^{l}}}{2\beta_j^{l}},
\\
\begin{aligned}
A_i^{\pm}&=\bigg(a_i^l-\sum_{r=1,r\neq
i}^nc_{ir}^Ml_r^{+}-\sum_{k=1}^md_{ik}^ll_{n+k}^{+}
 \pm\Big((a_i^l-\sum_{r=1,r\neq i}^nc_{ir}^Ml_r^{+}\\
&\quad -\sum_{k=1}^md_{ik}^ll_{n+k}^{+})^2
-4b_i^Mh_i^M\Big)^{1/2}\bigg)/(2b_i^M),
\end{aligned}
\\
A_{n+j}^{\pm}=\frac{\alpha_j^l-\sum_{r=1,r\neq j}^m
\gamma_{rj}^Ml_{n+r}^+
\pm\sqrt{(\alpha_j^l-\sum_{r=1,r\neq
j}^m\gamma_{rj}^Ml_{n+r}^+)^2-4\beta_j^Me_j^M}} {2\beta_j^M}
\end{gather*}
where
$i=1,2,\dots,n$; $j=1,2,\dots,m$.

In this paper, we use the following assumptions.

\begin{itemize}
\item[(H)] $a_i^l-\sum_{r=1,r\neq i}^nc_{ir}^Ml_r^{+}
-\sum_{k=1}^md_{ik}^ll_{n+k}^{+}>2\sqrt{b_i^{M}h_i^{M}}$,
$i=1,2,\dots,n$
and
$\alpha_j^l-\sum_{r=1,r\neq
j}^m\gamma_{rj}^Ml_{n+r}^+>2\sqrt{\beta_j^{M}e_j^{M}}$,
$j=1,2,\dots,m$.
\end{itemize}
By elementary calculus, one can easily show the following result.

\begin{lemma} \label{lem2.2}
Let $x>0$, $y>0$, $z>0$ and $x>2\sqrt{yz}$, for the functions
$ f(x,y,z)=\frac{x+\sqrt{x^{2}-4yz}}{2z}$ and
$ g(x,y,z)=\frac{x-\sqrt{x^{2}-4yz}}{2z}$, the
following assertions hold.
\begin{itemize}
\item[(1)] $f(x,y,z)$ and $g(x,y,z)$ are monotonically increasing
and monotonically decreasing on the variable $x\in(0,\infty)$,
respectively.
\item[(2)] $f(x,y,z)$ and $g(x,y,z)$ are monotonically decreasing
and monotonically increasing on the variable $y\in(0,\infty)$,
respectively.
\item[(3)] $f(x,y,z)$ and $g(x,y,z)$ are monotonically decreasing
and monotonically increasing on the variable $z\in(0,\infty)$,
respectively.
\end{itemize}
\end{lemma}

By  assumption (H) and Lemma 2.2, one can prove the following statement.

\begin{lemma} \label{lem2.3}
For the  equations
\begin{gather*}
a_i(t)-b_i(t)e^{u_i(t)}-h_i(t)e^{-u_i(t)}=0,\quad
\forall\, t\in\mathbb{R},\; i=1,2,\dots,n;
\\
\alpha_j(t)-\beta_j(t)e^{u_{n+j}(t)}-e_j(t)e^{-u_{n+j}(t)}=0,\quad
\forall \,  t\in\mathbb{R},\; j=1,2,\dots,m,
\end{gather*}
if assumption (H) holds, then we have the  inequalities
\begin{gather*}
\ln l_i^{-}<\ln u_i^{-}<\ln A_i^-<\ln A_i^+<\ln u_i^{+}<\ln
l_i^{+},  \quad \forall\,  t\in\mathbb{R}; \\
\ln l_{n+j}^{-}<\ln u_{n+j}^{-}<\ln A_{n+j}^-<\ln A_{n+j}^+<\ln
u_{n+j}^{+}<\ln l_{n+j}^{+}, \quad\forall\, t\in\mathbb{R},
\end{gather*}
where
\begin{gather*}
u_i^{\pm}=\frac{a_i(t)\pm\sqrt{(a_i(t))^{2}
-4b_i(t)h_i(t)}}{2b_i(t)},\quad i=1,2,\dots,n;\\
u_{n+j}^{\pm}=\frac{\alpha_j(t)\pm\sqrt{(\alpha_j(t))^{2}
-4\beta_j(t)e_j(t)}}{2\beta_j(t)},\quad
j=1,2,\dots,m.
\end{gather*}
\end{lemma}


\begin{theorem} \label{thm2.1}
Assume that {\rm (H)} holds. Then  \eqref{e1.1} has at
least $2^{n+m}$ positive
$\omega$-periodic solutions.
\end{theorem}

\begin{proof}
By making the substitutions
\begin{equation} \label{e2.1}
x_i(t)=\exp\{u_i(t)\},\quad
x_{n+j}(t)=\exp\{u_{n+j}(t)\},\quad i=1,2,\dots,n;\;j=1,2,\dots,m,
\end{equation}
 system \eqref{e1.1} can be reformulated as
\begin{equation} \label{e2.2}
\begin{gathered}
\begin{aligned}
 \dot{u}_i(t)&=a_i(t)-b_i(t)e^{u_i(t)}
-\sum_{r=1,r\neq i}^{n}c_{ir}(t)e^{u_{r}(t)}\\
 &\quad -\sum_{k=1}^{m}d_{ik}(t)e^{u_{n+k}(t)}-h_i(t)e^{-u_i(t)},
\end{aligned} \\
\begin{aligned} \dot{u}_{n+j}(t)
&=\alpha_j(t)-\beta_j(t)e^{u_{n+j}(t)}
-\sum_{r=1,r\neq j}^{m}\gamma_{rj}(t)e^{u_{n+r}(t)}\\
&\quad +\sum_{k=1}^{n}\delta_{kj}(t)e^{u_{k}(t)}-e_j(t)e^{-u_{n+j}(t)},
\end{aligned}
\end{gathered}
\end{equation}
where $i=1,2,\dots,n$; $j=1,2,\dots,m$.
Let
\[
X=Z=\big\{u=(u_{1},u_{2},\dots,u_{n+m})^T\in
C(R,R^{n+m}):u(t+\omega)=u(t)\big\}
\]
and define
\[
\|u\|=\sum_{i=1}^{n+m}\max_{t\in[0,\omega]}|u_i(t)|, \quad
 u\in X \mathrm{or }  Z.
\]
Equipped with the above norm $\|\cdot\|$, $X$ and $Z$ are Banach
spaces.  Let
\[
Lu=\dot{u}=\frac{\mathrm{d}u(t)}{\mathrm{d}t}
\]
and $N(u,\lambda)$ be column vector
{\scriptsize
\[
\begin{pmatrix}
a_{1}(t)-b_{1}(t)e^{u_{1}(t)}-\lambda\Big(\sum_{r=2}^{n}c_{1r}(t)e^{u_{r}(t)}
+\sum_{k=1}^{m}d_{1k}(t)e^{u_{n+k}(t)}\Big)-h_{1}(t)e^{-u_{1}(t)}\\
\vdots\\
a_{n}(t)-b_{n}(t)e^{u_{n}(t)}-\lambda\Big(\sum_{r=1}^{n-1}c_{nr}(t)e^{u_{r}(t)}
+\sum_{k=1}^{m}d_{nk}(t)e^{u_{n+k}(t)}\Big)-h_{n}(t)e^{-u_{n}(t)} \\
\alpha_{1}(t)-\beta_{1}(t)e^{u_{n+1}(t)}-\lambda\Big(\sum_{r=2}^{m}\gamma_{r1}(t)e^{u_{n+r}(t)}
-\sum_{k=1}^{n}\delta_{k1}(t)e^{u_{k}(t)}\Big)-e_{1}(t)e^{-u_{n+1}(t)}\\
\vdots\\
\alpha_{m}(t)-\beta_{m}(t)e^{u_{n+m}(t)}-\lambda\Big(\sum_{r=1}^{m-1}\gamma_{rm}(t)e^{u_{n+r}(t)}
-\sum_{k=1}^{n}\delta_{km}(t)e^{u_{k}(t)}\Big)-e_{m}(t)e^{-u_{n+m}(t)}\\
\end{pmatrix}
\]}
We put
$$
Pu=\frac{1}{\omega}\int_{0}^{\omega}u(t)\,\mathrm{d}t,\quad u\in X,\quad
Qz=\frac{1}{\omega}\int_{0}^{\omega}z(t)\,\mathrm{d}t,\quad z\in Z.
$$
Thus it follows that $\ker  L=R^{n+m}$,
$\operatorname{Im}L=\big\{z\in Z:\int_{0}^{\omega}z(t)\,\mathrm{d}t=0
\big\}$ is closed
in $Z$,
$\mathrm{dim}\ker  L=n+m=\operatorname{codim}\operatorname{Im} L$,
and $P,Q$ are continuous projectors such that
\[
\operatorname{Im} P=\ker  L,\quad  \ker  Q=\operatorname{Im}
L=\operatorname{Im} (I-Q).
\]
Hence, $L$ is a Fredholm mapping of index zero. Furthermore, the
generalized inverse (to $L$) $K_{P}:\operatorname{Im} L  \to
 \ker  P\bigcap \operatorname{Dom} L$ is given by
\[
K_{P}(z)=\int^{t}_{0}z(s)\,\mathrm{d}s
-\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{s}z(s)\,\mathrm{d}s.
\]
Then
\[
QN(u,\lambda)=\begin{pmatrix}
 \frac{1}{\omega}\int_{0}^{\omega}F_{1}(s,\lambda)\,\mathrm{d}s \\
     \vdots\\
 \frac{1}{\omega}\int_{0}^{\omega}F_{n+m}(s,\lambda)\,\mathrm{d}s\\
\end{pmatrix}_{(n+m)\times1}
\]
and
\begin{align*}
&K_{P}(I-Q)N(u,\lambda)\\
&=
 \begin{pmatrix}
 \int_{0}^{t}F_{1}(s,\lambda)\mathrm{d}s-
\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}F_{1}(s,\lambda)\,
\mathrm{d}s\mathrm{d}t
+(\frac{1}{2}-\frac{t}{\omega})\int_{0}^{\omega}F_{1}(s,\lambda)
\,\mathrm{d}s\\
   \vdots\\
 \int_{0}^{t}F_{n}(s,\lambda)\mathrm{d}s-
\frac{1}{\omega}\int_{0}^{\omega}\int_{0}^{t}F_{n}(s,\lambda)
\,\mathrm{d}s\mathrm{d}t
+ (\frac{1}{2}-\frac{t}{\omega})\int_{0}^{\omega}F_{n}(s,\lambda)
\,\mathrm{d}s
  \end{pmatrix}_{n\times1},
\end{align*}
where $F(u,\lambda)$ is the column vector
{\scriptsize \[
\begin{pmatrix}
a_{1}(s)-b_{1}(s)e^{u_{1}(s)}-\lambda\sum_{r=2}^{n}c_{1r}(t)e^{u_{r}(t)}
-\lambda\sum_{k=1}^{m}d_{1k}(s)e^{u_{n+k}(s)}-h_{1}(s)e^{-u_{1}(s)}\\
\vdots\\
a_{n}(s)-b_{n}(s)e^{u_{n}(s)}-\lambda\sum_{r=1}^{n-1}c_{nr}(s)e^{u_{r}(s)}
-\lambda\sum_{k=1}^{m}d_{nk}(s)e^{u_{n+k}(s)}-h_{n}(s)e^{-u_{n}(s)} \\
\alpha_{1}(s)-\beta_{1}(s)e^{u_{n+1}(s)}-\lambda\sum_{r=2}^{m}\gamma_{r1}(s)e^{u_{n+r}(s)}
+\lambda\sum_{k=1}^{n}\delta_{k1}(s)e^{u_{k}(s)}-e_{1}(s)e^{-u_{n+1}(s)}\\
\vdots\\
\alpha_{m}(s)-\beta_{m}(s)e^{u_{n+m}(s)}-\lambda\sum_{r=1}^{m-1}\gamma_{rm}(s)e^{u_{n+r}(s)}
+\lambda\sum_{k=1}^{n}\delta_{km}(s)e^{u_{k}(s)}-e_{m}(s)e^{-u_{n+m}(s)}\\
\end{pmatrix}.
\]}
 Obviously, $QN$ and $K_{P}(I-Q)N$ are
continuous. Using the Arzela-Ascoli theorem,
it is not difficult to show that
$K_{P}(I-Q)N(\overline{\Omega})$ is compact for any open bounded set
$\Omega\subset X$. Moreover,
$QN(\overline{\Omega})$ is clearly bounded. Thus, $N$ is $L$-compact
on $\overline{\Omega}$ with any open bounded set $\Omega\subset X$.

To use Lemma 2.1, we have to find at least $2^{n+m}$
appropriate open bounded subsets in $X$. Considering the operator
equation $Lu=\lambda N(u,\lambda),\lambda\in(0,1)$, we have
\begin{equation} \label{e2.3}
\begin{gathered}
\begin{aligned}
 \dot{u}_i(t)&= \lambda\Big(a_i(t)-b_i(t)e^{u_i(t)}
-\lambda\sum_{r=1,r\neq  i}^{n}c_{ir}(t)e^{u_{r}(t)}\\
&\quad -\lambda\sum_{k=1}^{m}d_{ik}(t)e^{u_{n+k}(t)}-h_i(t)
e^{-u_i(t)}\Big), \quad i=1,2,\dots,n,
\end{aligned}\\
\begin{aligned}
 \dot{ u}_{n+j}(t)
&= \lambda\Big(\alpha_j(t)-\beta_j(t)e^{u_{n+j}(t)}
  -\lambda\sum_{r=1,r\neq
  j}^{m}\gamma_{rj}(t)e^{u_{n+r}(t)}\\
 &\quad +\lambda\sum_{k=1}^{n}\delta_{kj}(t)e^{u_{k}(t)}
-e_j(t)e^{-u_{n+j}(t)}\Big),\quad j=1,2,\dots,m.
\end{aligned}
\end{gathered}
\end{equation}
 Assume that $u\in X$ is an $\omega$-periodic solution of
 \eqref{e2.3} for some $\lambda\in(0,1)$. Then there exist
$\xi_i,\eta_i,\xi_{n+j},\eta_{n+j}\in[0,\omega]$ such that
$u_i(\xi_i)=\max_{t\in[0,\omega]}u_i(t)$,
$u_i(\eta_i)=\min_{t\in[0,\omega]}u_i(t)$,
$u_{n+j}(\xi_{n+j})=\max_{t\in[0,\omega]}u_{n+j}(t)$,
$u_{n+j}(\eta_{n+j})=\min_{t\in[0,\omega]}u_{n+j}(t)$. It is
clear that
$\dot{u}_i(\xi_i)=0$,
$\dot{u}_i(\eta_i)=0$,
$\dot{u}_{n+j}(\xi_{n+j})=0$,
$\dot{u}_{n+j}(\eta_{n+j})=0$.
From this and \eqref{e2.3}, we have
\begin{equation} \label{e2.4}
\begin{gathered}
\begin{aligned}
&a_i(\xi_i)-b_i(\xi_i)e^{u_i(\xi_i)}-\lambda\sum_{r=1,r\neq
i}^{n}c_{ir}(\xi_i)e^{u_{r}(\xi_i)}\\
&-\lambda\sum_{k=1}^{m}d_{ik}(\xi_i)e^{u_{n+k}(\xi_i)}
-h_i(\xi_i)e^{-u_i(\xi_i)}=0,
\end{aligned} \\
\begin{aligned}
&\alpha_j(\xi_{n+j})-\beta_j(\xi_{n+j})e^{u_{n+j}(\xi_{n+j})}
-\lambda\sum_{r=1,r\neq j}^{m}\gamma_{rj}(t)e^{u_{n+r}(\xi_{n+j})}\\
&+\lambda\sum_{k=1}^{n}\delta_{kj}(\xi_{n+j})e^{u_{k}(\xi_{n+j})}
-e_j(\xi_{n+j})e^{-u_{n+j}(\xi_{n+j})}=0
\end{aligned}
\end{gathered}
\end{equation}
and
\begin{equation} \label{e2.5}
\begin{gathered}
\begin{aligned}
&a_i(\eta_i)-b_i(\eta_i)e^{u_i(\eta_i)}
-\lambda\sum_{r=1,r\neq i}^{n}c_{ir}(\eta_i)e^{u_{r}(\eta_i)}\\
&-\lambda\sum_{k=1}^{m}d_{ik}(\eta_i)e^{u_{n+k}(\eta_i)}
-h_i(\eta_i)e^{-u_i(\eta_i)}=0,
\end{aligned}  \\
\begin{aligned}
&\alpha_j(\eta_{n+j})-\beta_j(\eta_{n+j})e^{u_{n+j}(\eta_{n+j})}
-\lambda\sum_{r=1,r\neq j}^{m}\gamma_{rj}(t)e^{u_{n+r}(\eta_{n+j})}\\
&+\lambda\sum_{k=1}^{n}\delta_{kj}(\eta_{n+j})e^{u_{k}(\eta_{n+j})}
-e_j(\eta_{n+j})e^{-u_{n+j}(\eta_{n+j})}=0,
\end{aligned}
\end{gathered}
\end{equation}
where $i=1,2,\dots,n$; $j=1,2,\dots,m$. On the one hand, according to
the first equation of \eqref{e2.4}, we have
\begin{align*}
b_i^{l}e^{2u_i(\xi_i)}-a_i^{M}e^{u_i(\xi_i)}+h_i^{l}
&\leq b_i(\xi_i)e^{2u_i(\xi_i)}-a_i(\xi_i)e^{u_i(\xi_i)}
+h_i(\xi_i)\\&= -\lambda e^{u_i(\xi_i)}\Big(\sum_{r=1,r\neq
i}^{n}c_{ir}(\xi_i)e^{u_{r}(\xi_i)}+\sum_{k=1}^{m}
d_{ik}(\xi_i)e^{u_{n+k}(\xi_i)}\Big)\\
&< 0, \quad i=1,2,\dots,n;
\end{align*}
namely,
\[
b_i^{l}e^{2u_i(\xi_i)}-a_i^{M}
e^{u_i(\xi_i)}+h_i^{l}<0,\quad i=1,2,\dots,n,
\]
which implies
\begin{equation} \label{e2.6}
\ln l_i^{-}<u_i(\xi_i)<\ln l_i^{+},\quad i=1,2,\dots,n.
\end{equation}
 From this inequality and the second equation in \eqref{e2.4},
we obtain
\begin{align*}
&\beta_j^{l}e^{2u_{n+j}(\xi_{n+j})}
 -\alpha_j^{M}e^{u_{n+j}(\xi_{n+j})}+e_j^{l}\\
&\leq \beta_j(\xi_{n+j})e^{2u_{n+j}(\xi_{n+j})}
 -\alpha_j(\xi_{n+j})e^{u_{n+j}(\xi_{n+j})}
 +e_j(\xi_{n+j})\\
&= \lambda e^{u_{n+j}(\xi_{n+j})}\Big(-\sum_{r=1,r\neq
j}^{m}\gamma_{ir}(\xi_{n+j})e^{u_{n+r}(\xi_{n+j})}
+\sum_{k=1}^{n}\delta_{ik}(\xi_{n+j})e^{u_{k}(\xi_{n+j})}\Big)\\
&< e^{u_{n+j}(\xi_{n+j})}\sum_{k=1}^{n}\delta_{ik}^{M}l_{k}^{+},
\quad j=1,2,\dots,m;
\end{align*}
that is,
\[
\beta_j^{l}e^{2u_{n+j}(\xi_{n+j})}-\Big(\alpha_j^{M}
+\sum_{k=1}^{n}\delta_{ik}^{M}l_{k}^{+}\Big)e^{u_{n+j}(\xi_{n+j})}
+e_j^{l}<0,\quad j=1,2,\dots,m,
\]
which implies
\begin{equation} \label{e2.7}
\ln l_{n+j}^{-}<u_{n+j}(\xi_{n+j})<\ln
l_{n+j}^{+},\quad j=1,2,\dots,m.
\end{equation}
 From \eqref{e2.5}, we analogously have
\begin{gather} \label{e2.8}
\ln l_i^{-}<u_i(\eta_i)<\ln l_i^{+},\quad i=1,2,\dots,n \\
\label{e2.9}
\ln l_{n+j}^{-}<u_{n+j}(\eta_{n+j})<\ln l_{n+j}^{+},j=1,2,\dots,m.
\end{gather}
On the other hand, by \eqref{e2.4}, \eqref{e2.6} and \eqref{e2.7},
we obtain
\begin{align*}
a_i^l
&\leq a_i(\xi_i)\\
&=b_i(\xi_i)e^{u_i(\xi_i)}
 +\lambda\sum_{r=1,r\neq i}^{n}c_{ir}(\xi_i)e^{u_{r}(\xi_i)}
 +\lambda\sum_{k=1}^{m}d_{ik}(\xi_i)e^{u_{n+k}(\xi_i)}
 +h_i(\xi_i)e^{-u_i(\xi_i)}
\\
&< b_i^Me^{u_i(\xi_i)}+\sum_{r=1,r\neq
i}^{n}c_{ir}^Ml_r^++\sum_{k=1}^{m}d_{ik}^Ml_{n+k}^+
+h_i^Me^{-u_i(\xi_i)},\quad i=1,2,\dots,n,
\end{align*}
and
\begin{align*}
\alpha_j^l
&\leq \alpha_j(\xi_{n+j})=\beta_j(\xi_{n+j})
 e^{u_{n+j}(\xi_{n+j})}+\lambda\sum_{r=1,r\neq
j}^{m}\gamma_{rj}(t)e^{u_{n+r}(\xi_{n+j})}\\
&\quad -\lambda\sum_{k=1}^{n}\delta_{kj}(\xi_{n+j})
 e^{u_{k}(\xi_{n+j})}+e_j(\xi_{n+j})e^{-u_{n+j}(\xi_{n+j})}\\
&< \beta_j^Me^{u_{n+j}(\xi_{n+j})}+\sum_{r=1,r\neq
j}^{m}\gamma_{rj}^Ml_{n+r}^+
+e_j^Me^{-u_{n+j}(\xi_{n+j})},\quad j=1,2,\dots,m;
\end{align*}
namely,
\begin{gather*}
b_i^Me^{2u_i(\xi_i)}-\big(a_i^l-\sum_{r=1,r\neq
i}^{n}c_{ir}^Ml_r^+-\sum_{k=1}^{m}d_{ik}^Ml_{n+k}^+
\big)e^{u_i(\xi_i)}
+h_i^M>0,\quad i=1,2,\dots,n;
\\
\beta_j^Me^{2u_{n+j}(\xi_{n+j})}-\big(\alpha_j^l-\sum_{r=1,r\neq
j}^{m}\gamma_{rj}^Ml_{n+r}^+\big)e^{u_{n+j}(\xi_{n+j})}+e_j^M>0,
\quad j=1,2,\dots,m,
\end{gather*}
which implies
\begin{gather} \label{e2.10}
u_i(\xi_i)<\ln A_i^-\quad  \text{or}\quad
u_i(\xi_i)>\ln A_i^+,\quad i=1,2,\dots,n;\\
\label{e2.11}
u_{n+j}(\xi_{n+j})<\ln A_{n+j}^-\quad \text{or}\quad
u_{n+j}(\xi_{n+j})>\ln A_{n+j}^+,\quad j=1,2,\dots,m.
\end{gather}
According to \eqref{e2.5}, we analogously have
\begin{gather} \label{e2.12}
u_i(\eta_i)<\ln A_i^-\quad  \text{or}\quad
u_i(\eta_i)>\ln A_i^+,\quad i=1,2,\dots,n;\\
\label{e2.13}
u_{n+j}(\eta_{n+j})<\ln A_{n+j}^-\quad \text{or}\quad
u_{n+j}(\eta_{n+j})>\ln A_{n+j}^+,\quad j=1,2,\dots,m.
\end{gather}
 From \eqref{e2.6}-\eqref{e2.13} and Lemma 2.3,
we have that for any $t\in \mathbb{R}$,
\begin{gather} \label{e2.14}
\ln l_i^-<u_i(t)<\ln A_i^-\quad  \text{or} \quad
\ln A_i^+< u_i(t)<\ln l_i^+,\quad i=1,2,\dots,n;\\
 \label{e2.15}
\ln l_{n+j}^-<u_{n+j}(t)<\ln A_{n+j}^-\quad \text{or}\quad
\ln A_{n+j}^+<u_{n+j}(t)<\ln l_{n+j}^+,\quad j=1,2,\dots,m.
\end{gather}
For convenience, we denote
\[
G_i=\big(\ln l_i^{-},\ln A_i^-\big),\quad
H_i=\big(\ln A_i^+,\ln l_i^{+}\big),\quad i=1,2,\dots,n+m.
\]

Clearly, $ l_i^{\pm}$, $i=1,2,\dots,n+m$ are independent of
$\lambda$. For each $i=1,2,\dots,n+m$,  we choose one of  interval
among the two intervals $G_i$ and $H_i$ and denote it as $\Delta_i$,
then define the set
\[
\big\{u=(u_{1},u_{2},\dots,u_{n+m})^T\in X:u_i(t)\in \Delta_i,
t\in\mathbb{R}, i=1,2,\dots,n+m\big\}.
\]
Obviously, the number of the above sets is $2^{n+m}$. We denote
these sets as $\Omega_{k}$, $k=1,2,\dots,2^{n+m}$.
$\Omega_{k}, k=1,2,\dots,2^{n+m}$ are bounded open subsets of
$X, \Omega_i\cap\Omega_j=\phi,i\neq j$. Thus
$\Omega_{k}$ $(k=1,2,\dots,2^{n+m})$ satisfies the requirement (a)
in Lemma 2.1.

Now we show that (b) of Lemma 2.1 holds; i.e., we prove when
$u\in\partial\Omega_{k}\cap \ker  L=\partial\Omega_{k}\cap
R^{n+m},QN(u,0)\neq(0,0,\dots,0)^{T}$,
$k=1,2,\dots,2^{n+m}$. If it is not true, then when
$u\in\partial\Omega_{k}\cap \ker L=\partial\Omega_{k}\cap R^{n+m}$,
$i=1,2,\dots,2^{n+m}$, constant
vector $u=(u_{1},u_{2},\dots,u_{n+m})^{T}$ with
$u\in\partial\Omega_{k},k=1,2,\dots,2^{n+m}$, satisfies
\begin{gather*}
    \int_{0}^{\omega}a_i(t)\,\mathrm{d}t
-\int_{0}^{\omega}b_i(t)e^{u_i}\,\mathrm{d}t
    -\int_{0}^{\omega}h_i(t)e^{-u_i}\,\mathrm{d}t=0,\quad
i=1,2,\dots,n;\\
 \int_{0}^{\omega}\alpha_j(t)\,\mathrm{d}t
-\int_{0}^{\omega}\beta_j(t)e^{u_{n+j}}\,\mathrm{d}t
    -\int_{0}^{\omega}e_j(t)e^{-u_{n+j}}\,\mathrm{d}t=0,\quad
j=1,2,\dots,m.
\end{gather*}
In view of the mean value theorem of calculous, there exist $n+m$
points $t_i$ $(i=1,2,\dots,n+m)$ such that
\begin{gather} \label{e2.16}
a_i(t_i)-b_i(t_i)e^{u_i}-h_i(t_i)e^{-u_i}=0,
\quad i=1,2,\dots,n;\\
 \label{e2.17}
\alpha_j(t_{n+j})-\beta_j(t_{n+j})e^{u_{n+j}}
-e_j(t_{n+j})e^{-u_{n+j}}=0,\,\,j=1,2,\dots,m.
\end{gather}
By \eqref{e2.16} and \eqref{e2.17}, we have
\begin{gather*}
u_i^{\pm}=\frac{a_i(t_i)\pm\sqrt{(a_i(t_i))^{2}
-4b_i(t_i)h_i(t_i)}}{2b_i(t_i)},\quad i=1,2,\dots,n;
\\
u_{n+j}^{\pm}=\frac{\alpha_j(t_{n+j})
\pm\sqrt{(\alpha_j(t_{n+j}))^{2}
-4\beta_j(t_{n+j})e_j(t_{n+j})}}{2\beta_j(t_{n+j})},
\quad j=1,2,\dots,m.
\end{gather*}
According to Lemma 2.3, we obtain
\[
\ln l_i^{-}<\ln u_i^{-}<\ln A_i^-<\ln A_i^+<\ln u_i^{+}<\ln
l_i^{+},\quad i=1,2,\dots,n+m.
\]
Then $u$ belongs to one of
$\Omega_{k}\cap R^{n+m}$, $k=1,2,\dots,2^{n+m}$.
This contradicts the fact that
$u\in\partial\Omega_{k}\cap R^{n+m}$, $k=1,2,\dots,2^{n+m}$. This
proves (b) in Lemma 2.1 holds.

Finally, we show that (c) in Lemma 2.1 holds.  Since $(H)$ holds,
the system of algebraic equations
\begin{gather*}
a_i(t_i)-b_i(t_i)e^{x_i}-h_i(t_i)e^{-x_i}=0,\quad
i=1,2,\dots,n,\\
\alpha_j(t_{n+j})-\beta_j(t_{n+j})e^{x_{n+j}}
-e_j(t_{n+j})e^{-x_{n+j}}=0,\quad j=1,2,\dots,m
\end{gather*}
 has $2^{n+m}$ distinct solutions:
\[
(x_{1}^{*},x_{2}^{*},\dots,x_{n+m}^{*})
=(\ln \hat{x}_{1},\ln\hat{x}_{2},\dots,\ln\hat{x}_{n+m}),
\]
where $\hat{x}_i=x_i^{-}$ or
$\hat{x}_i=x_i^{+}$,
\[
x_i^{\pm}=\frac{a_i(t_i)\pm\sqrt{(a_i(t_i))^{2}
-4b_i(t_i)h_i(t_i)}}{2b_i(t_i)},\quad
i=1,2,\dots,n
\]
and $ \hat{x}_{n+j}=x_{n+j}^{-}$ or $\hat{x}_{n+j}=x_{n+j}^{+}$,
\[
x_{n+j}^{\pm}
=\frac{\alpha_j(t_{n+j})\pm\sqrt{(\alpha_j(t_{n+j}))^{2}
-4\beta_j(t_{n+j})e_j(t_{n+j})}}{2\beta_j(t_{n+j})},\quad
j=1,2,\dots,m.
\]
By Lemma 2.2, it is easy to verify that
\[
\ln l_i^{-}<\ln x_i^{-}< \ln A_i^-<\ln A_i^+<\ln x_i^{+}<\ln
l_i^{+},\,\,i=1,2,\dots,n+m.
\]

Therefore, $(x_{1}^{*},x_{2}^{*},\dots,x_{n+m}^{*})$
belongs to the corresponding $\Omega_{k}$. Since
$\ker L=\operatorname{Im}Q$, we can take $J=I$.
A direct computation gives, for $k=1,2,\dots,2^{n+m}$,
\begin{align*}
&\deg\big\{JQN(u,0),\Omega_{k}\cap \ker L,(0,0,\dots,0)^{T}\big\}\\
&= \operatorname{sign}\Big[\prod_{i=1}^{n}\prod_{j=1}^{m}
\Big(-b_i(t_i)x_i^{*}+\frac{h_i(t_i)}{x_i^{*}}\Big)
\Big(-\beta_j(t_{n+j})x_{n+j}^{*}
+\frac{e_j(t_{n+j})}{x_{n+j}^{*}}\Big)\Big].
\end{align*}
Since
\begin{gather*}
a_i(t_i)-b_i(t_i)x_i^{*}-\frac{h_i(t_i)}{x_i^{*}}=0,
\quad i=1,2,\dots,n,\\
\alpha_j(t_{n+j})-\beta_j(t_{n+j})x_{n+j}^{*}
-\frac{e_j(t_{n+j})}{x_{n+j}^{*}}=0,\quad j=1,2,\dots,m,
\end{gather*}
it follows that
\begin{align*}
&\deg\big\{JQN(u,0),\Omega_{k}\cap \ker L,(0,0,\dots,0)^{T}\big\}\\
&= \operatorname{sign}\Big[\prod_{i=1}^{n}
 \prod_{j=1}^{m}\big(a_i(t_i)-2b_i(t_i)x_i^{*}\big)
\big(\alpha_j(t_{n+j})-2\beta_j(t_{n+j})x_{n+j}^{*}\big)\Big]\\
&= \pm 1, \quad k=1,2,\dots,2^{n+m}.
\end{align*}

So far, we have proved that $\Omega_{k}$ $(k=1,2,\dots,2^{n+m})$
satisfies all the assumptions in Lemma 2.1.
Hence, system \eqref{e2.2} has
at least $2^{n+m}$ different $\omega$-periodic solutions.
Thus by \eqref{e2.1} system \eqref{e1.1} has at least $2^{n+m}$
different positive $\omega$-periodic solutions.
This completes the proof of Theorem 2.1.
\end{proof}

For  system \eqref{e1.1}, assume that $c_{ir}(t)\geq0$
$(i=1,2,\dots,n;r\neq i)$,
$\gamma_{rj}(t)\geq0$ $(j=1,2,\dots,m;r\neq j)$,
$d_{ik}(t)\geq0$ $(i=1,2,\dots,n;k=1,2,\dots,m)$ and
$\delta_{kj}(t)\geq0$ $(j=1,2,\dots,m;k=1,2,\dots,n)$ and
$a_i(t)>0$, $b_i(t)>0$, $h_i(t)>0$, $\alpha_j(t)>0$,
$\beta_j(t)>0$, $e_j(t)>0$
are continuous periodic functions, similar to the proof
of Theorem 2.1, one can prove the following result.

\begin{theorem} \label{thm2.2}
Assume that (H) hold. Then  \eqref{e1.1} has at least $2^{n+m}$
positive $\omega$-periodic solutions.
\end{theorem}


\section{Example}

 Now, let us consider the following
network-like predator-prey system with harvesting terms which have
one prey species ($n=1$) and two predator species ($m=2$):
\begin{equation} \label{e3.1}
  \begin{gathered}
    \dot{x}_{1}(t)= x_{1}(t)\Big(3+\sin t-\frac{4+\sin
    t}{10}x_{1}(t)-d_{12}(t)x_{2}(t)-d_{13}(t)x_{3}(t)
    \Big)-\frac{9+\cos t}{20}, \\
    \dot{x}_{2}(t)= x_{2}(t)\Big(3+\cos t-\frac{5+\cos t}{10}x_{2}(t)+\delta_{11}(t)x_{1}(t)
    -\gamma_{21}(t)x_{3}(t)\Big)-\frac{2+\cos t}{5},\\
    \dot{x}_{3}(t)= x_{3}(t)\Big(3+\sin 2t-\frac{8+\sin 2t}{10}x_{3}(t)
    +\delta_{12}(t)x_{1}(t)-\gamma_{12}(t)x_{2}(t)\Big)-\frac{8+\cos 2t}{10}.
  \end{gathered}
\end{equation}
In this case, $a_{1}(t)=3+\sin t$,
$b_{1}(t)=\frac{4+\sin t}{10}$,
$h_{1}(t)=\frac{9+\cos t}{20}$,
$\alpha_{1}(t)=3+\cos t$,
$\beta_{1}(t)=\frac{5+\cos t}{10}$,
$e_{1}(t)=\frac{2+\cos t}{5}$,
$\alpha_{2}(t)=3+\sin 2t$,
$\beta_{2}(t)=\frac{8+\sin 2t}{10}$,
$e_{2}(t)=\frac{8+\cos 2t}{10}$.
Since
\[
l_1^{\pm}=\frac{a_1^M\pm\sqrt{(a_1^M)^2-4b_1^lh_1^l}}{2b_1^l}
=5\pm\frac{1}{2}\sqrt{97},
\]
taking $\delta_{11}(t)=\delta_{11}(t+2\pi)>0$,
$\delta_{12}(t)=\delta_{12}(t+2\pi)>0$ such that
$\delta_{11}^Ml_1^+=\delta_{12}^Ml_1^+=1$,
then we have
\[
l_3^{\pm}=\frac{(\alpha_2^M+\delta_{12}^Ml_1^+)
\pm\sqrt{(\alpha_2^M+\delta_{12}l_1^+)^2-4\beta_2^le_2^l}}
{2\beta_2^l}=\frac{25\pm\sqrt{576}}{7}.
\]
Take $d_{12}(t)=d_{12}(t+2\pi)>0,d_{13}(t)
=d_{13}(t+2\pi)>0,\gamma_{21}(t)
=\gamma_{21}(t+2\pi)>0,\gamma_{12}(t)
=\gamma_{12}(t+2\pi)>0$
such that
$d_{11}^ll_2^+=d_{12}^ll_3^+=\gamma_{21}^Ml_3^+
=\gamma_{12}^Ml_2^+=\frac{1}{10}$,
then
$1.8=a_{1}^{l}-d_{11}^ll_2^+-d_{12}^ll_3^+
>2\sqrt{b_{1}^{M}h_{1}^{M}}=1$,
$1.9=\alpha_{1}^{l}-\gamma_{21}^Ml_3^+>2\sqrt{\beta_{1}^{M}e_{1}^{M}}
=\frac{6}{5}$,
$1.9=\alpha_{2}^{l}-\gamma_{12}^Ml_2^+
>2\sqrt{\beta_{2}^{M}e_{2}^{M}}=\frac{18}{10}$.
Therefore, all conditions of Theorem 2.1 are satisfied.
By Theorem 2.1, system \eqref{e3.1} has at least eight
positive $2\pi$-periodic solutions.


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\end{document}